An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

作者：Yu, Pei^*; Han, Maoan; Li, Jibin

来源：International Journal of Bifurcation and Chaos, 2018, 28(6): 1850078.

DOI：10.1142/S0218127418500785

• Homogeneous polynomial system
• nondegenerate center
• Hopf bifurcation
• limit cycle
• normal form

In the two articles in Appl. Math. Comput., J. Gine [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: (x) over dot = -y + Pn(()x, y), (y) over dot = x + Q(n)(x, y), where P-n and Q(n) are homogeneous polynomials of degree n. It is shown that the maximal number of the small limit cycles, denoted by M-h(n), satisfies M-h(n) >= 2n - 1 for n = 4, 5, 6, 7; and M-h(8) >= 13, M-h(9) >= 16. It seems that the correct answer for their case n = 9 should be M-h(9) >= 15. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that M-h(n) >= 2n for n = 4, 5, 6, 7; and M-h(n) >= 2(n - 1) for n = 8, 9, which improve Gine's results with one more limit cycle for each case.

• 出版日期2018-6-15
• 单位华侨大学; 曲阜师范大学; 上海师范大学